Validation of traffic models using PTV VISSIM 7

Transcription

1 Department of Mechanical Engineering Manufacturing Networks Research Group Validation of traffic models using PTV VISSIM 7 J.H.T. Hendriks Supervisors: Dr.ir. A.A.J. Lefeber Ir. S.T.G. Fleuren Eindhoven, April 2015

2 Table of contents Terminology 2 1 Introduction 3 2 Overview of literature Webster s equation Miller s equation Van den Broek s equation Other equations VISSIM Settings VISSIM s limits Simulation results Inter-arrival time Departure rate for different directions Diffusion over long distances Conclusion 16 6 Recommendations 17 References 18 TU/e 1

3 Terminology In this section we explain the used terminology for this report: Fixed-time traffic signal Intersection Queue length Mean waiting time Arrival rate Maximum arrival rate Inter-arrival time Departure rate Maximum departure rate Cycle time Effective green-time Effective red-time Traffic lights cycling in a fixed sequence of red and green lights. Junction with a fixed-time traffic-signal. Amount of cars waiting at an intersection. Average time E[D] a vehicle has to wait at an intersection in [s]. The average rate λ Arrival rate of vehicles at an intersection in [cars/s]. Maximum possible arrival rate in [cars/h] for the set speed-limit. Time between the arrival of two successive cars in a simulation in [s]. Departure rate µ of vehicles at an intersection in [cars/s]. The average rate with which traffic departs when the maximum queue is non-empty. Cycle time c of a traffic signal in [s]. In one cycle, all traffic lights of an intersection have been green at least once. Effective green time g of a traffic signal in [s], the time whereby vehicles are allowed to leave the queue. The time whereby vehicles are not allowed to leave the queue in [s]. Could also be expressed as c g. TU/e 2

4 1 Introduction Road transportation has always been an important method of travel. Over the last century, mobility has grown rapidly with a increasing demand for a higher travel distance. The need to travel longer distances also caused an increasing amount of road traffic over the years [1]. However, this increase in traffic also caused an increase of unwanted traffic congestions, which have a large negative impact on the average travel speed. To keep congestions at intersections at a minimum, one has to make thoughtful decisions on the scheduling of traffic lights: how long should the traffic lights stay green and how long should they be red, so that the amount of waiting vehicles stays at a minimum. The main goal of this report is to validate if an M/D/1-system is a reasonable assumption for traffic intersections, since traffic models are often based on this system. The validation is done using the traffic simulator PTV VISSIM 7. VISSIM is a microscopic traffic simulator, meaning that it simulates each vehicle individually. With VISSIM, it is possible to determine the distribution of the arrival rate λ and the departure rate µ of the road or intersection. Furthermore, one could also determine the diffusion and queues in a network of intersections or over longer distances using this simulator. VISSIM s settings and properties are based on real measurements performed in Germany [2]. Before we go to the simulations, we first look for existing formulas for the mean waiting time which are based on an M/D/1-system. This is done in Section 2. These formulas depend on the arrival rate λ, departure rate µ and the amount of red and green time of the traffic lights. To keep the simulations reliable and reproducible, we discuss all relevant settings, assumptions and the functionality of VISSIM in Section 3. After that, we move to the simulation results in Section 4. The simulations measure the distribution of the arrival rate and the departure rate, so that the assumption of an M/D/1-system in traffic is validated. Furthermore, the diffusion and congestions over longer distances is analysed. The analysing is done by simulating the effect of long distances on the mean time between each car and the mean distance between each car. Since a lot of literature about traffic models are based on an M/D/1-system, it is important to validate this via simulations, which is the main contribution of this report. So this report determines whether the arrival rate at a single intersection is exponential or not. It also determines the different possible departure rates at an intersection, using a maximum enter arrival rate and different speed limits due to the curves in the road. Furthermore, this report determines the diffusion over long distances. Thus the report also looks at the effect of so-called ghost traffic jams, traffic jams that appear due to sudden braking and speed differences. TU/e 3

5 2 Overview of literature Since the goal of this report is to validate whether an M/D/1-system is a reasonable assumption for traffic intersections, we first need to look at already published approximations for the mean waiting time at a traffic light. These approximations are a function of: ˆ λ = arrival rate in [cars/s] ˆ µ = departure rate in [cars/s] ˆ c = cycle time in [s] ˆ g = effective green time in [s] ˆ ρ = λ µ These terms are explained in more detail in the Terminology section. In this section we look at three different approximation formulas of Webster [3], Miller [3] and van den Broek [4]. After that, the assumptions on which these approaches are based are validated in Section 4. The choice of Webster s and Miller s formulas is based on the common use of these approximations in the past. These formulas are one of the earliest approximations about the mean waiting time at an intersection using an M/D/1-system, with Miller s paper only five years older than Webster s. Van den broek s formula is one of the fairly recent approximations. 2.1 Webster s equation The first equation is the oldest of the three equations, dating from Webster s approximation of the mean waiting time is [5]: E[D] = (c g)2 2c(1 ρ) + ρc 2 2g(µg λc) 0.65( c λ 2 )2 ( λc µg )2+5g/c. (2.1) This equation is based on an M/D/1-model and is compared with observations of London traffic. While the first two terms are determined theorically, the third term is a correction term based on simulation results [6]. 2.2 Miller s equation The second equation is from Miller [3]. We formulate this equation as was done in [6] and [4]: E[D] = c g 2c(1 ρ) + [(c g) + E[XBR ] + 1 λ µ (1 + 1 )], (2.2) 1 ρ where E[X BR ] is the number of waiting vehicles at the beginning of effective red time of the traffic signal, where E[X BR ] equals E[X BR ] = exp[ 1.33 µg(1 ρ )/ρ ] 2(1 ρ, (2.3) ) TU/e 4

6 where ρ = λc µg. (2.4) Equation (2.2) is based on results observed on several junctions in Birmingham. 2.3 Van den Broek s equation The last equation is from Van den Broek [6]: E[D] = l λ + (c g)2 2c(1 ρ) + (ρ ) 4 c g 2(1 ρ)(µg λc), (2.5) where l is the amount of vehicles that remain after the traffic signal goes from green to red light. l is given by: l = ρ + ρ 2(1 ρ). (2.6) Note that the third term of Equation Equation (2.5) is similar to the first term of Webster s equation, which is based on a theoretical consideration for a deterministic arrival rate, whereby the second term compensates for the stochastic behaviour of the arrival rate. This however, is noted differently in (2.5), setting different equations for E[X BR ] and a seperate term for a starting queue after green time. Van den Broek also states that (2.5) yields more accurate results than the approximations of Webster and Miller. 2.4 Other equations There are several more approximations regarding the mean waiting time at an intersection. While we won t name them all in this section, we name a few of them as example: Clayton [7] is one of the oldest approximations, even older than Webster s. However, the pattern of the arrival rate in Clayton s model differs from the other formulas. Rendering it unusable for an M/D/1-system. Winsten [8] used a binomial distribution for the arrival rate. The approximation of Winsten is slightly more accurate to practice when compared to Clayton s, but it is still not as accurate as the formulas of Webster, Newell [3] and van den Broek. Newell [9] developed the formula two years after Miller. However, since Miller s approximation yields better results and was used by van den Broek [6], we chose Miller s over Newell s formula. Akçelik [10] created an approximation that is also useful for overflowing queue lengths, since the formulas of Webster and Miller only assume steady-state conditions. Meaning that Akçelik s formula also takes the possibility of oversaturation into account. Now that the formulas about traffic queues in an M/D/1-system are discussed, we can validate the distribution of the arrival rate and the departure rate in these approximations using VISSIM. In the next section we discuss the relevant settings and functionalities of VISSIM. TU/e 5

7 3 VISSIM In the previous section we discussed relevant approximations for the mean waiting time at an intersection with traffic signals. Before we go to the simulations, we need to discuss all relevant settings, assumptions and the functionality of the simulation program, so that the simulations are reliable and reproducable. For the simulations we use PTV VISSIM 7, which is a microscopic traffic simulator developed by PTV Planung Transport Verkehr AG [2]. Furthermore, during the simulations VISSIM appears to have some limitations and uncertainties which are discussed at the end of this section. Once all relevant settings and limitations are known, we can validate the distribution of the arrival rate and the departure rate of the formulas of Section 2, which is done in the next section. 3.1 Settings The first relevant setting we discuss is the way how traffic density is defined. There are two different settings for the volume type of the vehicle input: exact and stochastic. For exact, the specified traffic volume will be reached exactly by the end of the simulation, though the inter-arrival time remains stochastic. For the other setting, stochastic fluctuations of the specified traffic volume may occur. The expectation is that the inter-arrival time has an exponential distribution. This is validated in Section 4.1 for both exact and stochastic. The default curves of the acceleration and deceleration of the vehicles are based on [11]. This traffic flow model is based on measurements in Germany in 1974 and assumes four different driving states: ˆ Free driving: Speed oscillates around the desired speed while there is no influence of other traffic. ˆ Approaching: Adapting its speed to the lower speed of the preceding vehicle, until it reaches its desired safety distance. ˆ Following: The difference in speed between its preceding car oscillates around zero. ˆ Braking: Medium to high deceleration if the safety distance becomes violated. All simulations are done with these default acceleration and deceleration settings, unless specified otherwise. Furthermore, all cars have a desired speed between 48 and 58 km/h, so in all simulations the roads are assumed to be areas with a maximum speed of 50 km/h. The desired speed is uniformly distributed over all vehicles in all simulations, unless specified otherwise. So in the end the average speed is 53 km/h. VISSIM assumes that vehicles tend to exceed the speed limit very slightly. Other relevant settings that are used for the simulations: ˆ Only cars are used during the simulation. Effects of busses, trucks and other type of vehicles are neglected. ˆ The vehicle inputs of all simulations are run on exact, unless specified otherwise. ˆ All simulations have been carried out 30 times, with each simulation having a simulation time of 600s, which is the maximum allowed simulation time for the student version of VISSIM. ˆ The step time of all simulations is 0.1s, meaning that VISSIM calculates the new situation for all cars every 0.1s. TU/e 6

8 3.2 VISSIM s limits Although PTV VISSIM 7 has a wide variety of options, there are some limits in terms of accuracy. One of the inaccuraties of VISSIM is the distribution of the inter-arrival time. While the time step of VISSIM is 0.1s, the vehicle inputs of VISSIM have a step time of roughly 0.5s. Due to this, it is necessary to round off all collected data times to 0.5s. This issue is discussed in more detail in Section 4.1. For followup simulations, one should keep in mind that VISSIM is not capable of spawning vehicles with the exact same speed, if wanted. VISSIM is only capable to spawn vehicles within a specified speed distribution. It is possible to set the spawning speed difference very low. The lowest speed difference possible is 0.01 m/s. Furthermore, it is advisable to avoid nodes as a data collection function within VISSIM. While you can define certain areas to collect a large variety of data using nodes, the nodes tend to skip vehicles during the collection of the data. This reduces the reliability of the data. Instead, the data collection for this report is done via vehicle input data, vehicle record and data collection (raw data) points. Now that the settings and limitations of VISSIM are known, we can continue with the simulations. This is done in the next section. TU/e 7

9 4 Simulation results Now that we discussed the relevant settings and functionalities of VISSIM, we validate in this section the distribution of the inter-arrival time and the departure rate of the formulas in Section 2. Since the formulas are based on an M/D/1-system, the approxmimations assume an inter-arrival time with an exponential distribution. After that, we determine the departure rates at an intersection in Section 4.2. This is done with a maximum arrival rate and different directions at an intersection, with each direction having its own speed limit. In Section 4.3, we analyse the effect of long distances on the distribution and diffusion of the time and distance in between cars. For this analysis there is a simulation done for both the maximum possible arrival rate and a moderate arrival rate. 4.1 Inter-arrival time To validate that the inter-arrival time has an exponential distribution in VISSIM, the time between each car entering the simulation is measured. There are 30 simulations for both exact and stochastic, of which the results are compared. The inserted arrival rate for this simulation is 900 cars/h. Figure 4.1: Inter-arrival time distribution withoutfigure 4.2: Inter-arrival time distribution with rounding off rounding off to 0.5s Figure 4.1 shows the distribution of the inter-arrival time for all simulations together. The first thing that is noticable are peaks on a constant interval of 0.5s, which is not a realistic occurence. The VISSIM manual does not mention these types of intervals for the vehicle input [2]. Nor is it possible that car speed and acceleration influences the inter-arrival time, since the measuring occurs on the moment that a car enters the simulation. A possibility is that the vehicle inputs spawn cars in an inter-arrival time of an interval of 0.5s, which causes a peak in the same interval. The amount of cars that each interval spawns decreases exponentially. Still this does not explain the small amount of cars beside the peaks. A possible explanation is that VISSIM creates a normal distribution within each 0.5s-interval. However, this all is still a hypothesis that has yet to be confirmed. One should round off to 0.5s for all simulations to overcome this uncertainty. Figure 4.2 visualizes the inter-arrival time distribution when rounded off to 0.5s. This time, the expected exponential distibution is visible. To valide the hypothesis of a exponential distribution, one could carry out a Lilliefors test in MATLAB. The results are shown in Table 4.1. In this table, k represents the test statistic of the plot, of which the value depends on the mean and variance compared to the expected distribution. In this section the expected distribution is exponen- TU/e 8

10 tial. The critical value of the plot c crit depends on the amount of data points in the plot. Since the critical value c is in both cases higher than the test statistic k, the null hypothesis for a exponential distribution is valid, using a significance level of 95%. Table 4.1: Results of a Lilliefors test for Figure 4.2 k c crit exact stochastic Figures 4.3 and 4.4 show the average and variance of the inter-arrival time for each simulation. As expected, the average for exact is almost always 4.0s, since this vehicle input always generates 150 cars per simulation. Since this is not the case for stochastic, a flatter normal distribution is visible. However, we want to know if this flatter normal distribution could be still considered a normal distribution. Figure 4.3: Average inter-arrival time for each simu-figurlation 4.4: Variance of the inter-arrival time for each simulation Testing the null hypothese for a normal distribution in figures 4.3 and 4.4 is done with a Lilliefors test in MATLAB. The results are displayed in Table 4.2. Table 4.2: Results of a Lilliefors test for figures 4.3 and 4.4 k c crit variance for exact variance for stochastic average for exact average for stochastic Apart from the average plot for stochastic, the critical value is in all cases higher than the test statistic, which means that the null hypotheses for a normal distribution is valid for the three plots, using a significance level of 95%. Table 4.3 represents the coefficient of variation of the inter-arrival times, together with the relevant average standard deviation σ and the average of all 30 simulations. As expected, the average interarrival time for both exact and stochastic is close to 4.0 seconds. According to the coefficient of variation, both settings are exponential. TU/e 9

11 Table 4.3: Coefficient of variation of the inter-arrival time for exact and stochastic σ [s] average [s] c v Exact Stochastic Now that we have verified an exponential distribution for the inter-arrival time, we verify a deterministic distribution for the departure rate in the next section. 4.2 Departure rate for different directions In this section we determine the departure rate for every direction at an intersection. This simulation shows the impact of the curve length and its speed limit on the departure rate. However, since VISSIM does not take reduction in speed for sharp turns into account, it is necessary to manually add the different speed limits for every direction. For the left turn we use a speed limit of 25 and 30 km/h, while for the right turn we use 15 and 20 km/h. Figure 4.5 shows the simulation that is modelled in VISSIM. Figure 4.5: Modelled simulation within VISSIM for calculating the departure rates To guarantee a steady-state of incoming cars over the detectors on the junctions, the first 80 seconds of the simulation time are not taken into account with the results. Red traffic lights are added for the first 60 seconds to create queues for a sufficient stream of cars. The 20 seconds of simulation time after red light are neglected due to the accelerations of the cars. Table 4.4: Departure rate for different directions speed limit [km/h] µ [cars/h] σ [cars/h] c v Straight left left right right Table 4.4 shows the departure rate for different directions on a junction, together with the standard deviation and the coefficient of variation. As expected, the departure rate for right is the lowest and for a straight line the highest, since curves generate a certain speed limit in terms of safety. And since TU/e 10

12 a right turn is a sharp curve, the speed limit is the lowest, which has a big impact on the departure rate. Furthermore, the departure rate of 2618 cars/h for a straight line also equals the arrival rate for this simulation and for Section 4.3, since the inserted arrival rate is set on maximum for a speed limit of 50 km/h. The standard deviations are very low and the coefficient of variations are almost zero. This means the distribution of the departure rate is deterministic. 4.3 Diffusion over long distances In this section we determine the effect of long distances on the results, before the traffic reaches an intersection with traffic lights. First we determine te distribution for the time between each car at a high arrival rate. After that we visualize the diffusion of the cars over a long distance. This vizualisation is done for a moderate arrival rate. High arrival rate This simulation visualises the effect of distance on the time between each car, especially due to congestions like ghost traffic jams with a high arrival rate. These traffic jams are caused by speed differences or sudden braking. A hypothesis is that the time between each car increases with a longer travel distance. While this section focusses more on delays due to heavy traffic congestions over a long road, Section 4.3 focusses on the diffusion of vehicles over a long distance with a lower arrival rate. To make sure to get a maximum amount of ghost traffic jams, the arrival rate is set on maximum, which is effectively about 3000 cars/h. All results are rounded off to 0.5s as discussed in Section 4.1. Figure 4.6: Time between each car on several distances Figure 4.6 shows the time between each car on several distances within the simulation. The amount of measured cars is about 9000 to for each distance. The expected increase of time between each car does not occur, it stays relatively the same. Ghost traffic jams occur on every distance of the simulation due to the high traffic volume, thus the effect of these traffic jams is the same for all distances. The shape of the plots represent a gamma distribution. As comparison a gamma distribution is added with α = 22.5 and β = TU/e 11

13 Figure 4.7: Average time between each car for eachfigure 4.8: Variance of time between each car for simulation each simulation Figures 4.7 and 4.8 show the average and variance of the time between each car for each simulation. The average time does seem to increase over an increasing distance, though at a minimum rate. This slightly increasing average time is also visible in Table 4.5. Table 4.5: Coefficient of variation of the time between each car distance [m] σ [s] average [s] c v However, the variance and thus the standard variation seems to vary heavily without any specific order, resulting in a fluctuating coefficient of variation c v. Expected is that the plots and variance become more constant for an increased amount of simulations. A possible cause of this fluctuation is that the amount of ghost traffic jams on each distance differs in each simulation. Furthermore, there is a slight difference of height in the peaks of Figure 4.6, which could also be caused by the different amount of ghost traffic jams. Despite the fluctuating coefficient of variation, all distances remain clearly low variant. Table 4.6 shows the relevant results of a Lilliefors test in MATLAB for a normal distribution. The criticle value c crit is in all cases , which means the null hypothesis for all plots in figures 4.7 and 4.8. Table 4.6: Results of a Lilliefors test for figures 4.7 and 4.8 distance [m] k average k variance TU/e 12

14 Moderate arrival rate In this simulation we want to visualize the diffusion over a long distance. To simulate the average distance between each car over time, one uses a long road on which the cars are at least 175 seconds present in the simulation. The goal is to see if time influences the average distance between cars. This is also done for multiple cars in between. To prevent that some cars have more influence on the results than other cars, every distance between each car is only used once. This is done for each amount of cars in between on each timestep. So for the measurement of two cars in between, we calculate the distance between car number 1 and 4, after that we do the same for car number 4 and 7. The arrival rate is set on 900 cars/h, which is equal to the simulations of Section 4.1. This arrival rate is high enough for a sufficient amount of cars in the simulation, but low enough to keep ghost traffic jams at a minimum, since the goal of this simulation is not to examine the effect of ghost traffic jams. Figure 4.9: Average distance between cars as function of time Figure 4.9 shows the average distance between cars over the time present in the simulation. During the simulations, the distance of some cars with its predecessor become bigger. However, this also means that the distance with the car in the back becomes smaller, since the average speed of all cars together is 53 km/h. This results in a constant overall distance between the cars, which is as expected with the VISSIM settings explained in Section 3.1. Looking at multiple cars in between gives similar results. The standard deviation however lineary increases over time, as shown in Figure The distance between cars varies more as they progress in the simulation, which results in an increasing standard deviation. TU/e 13

15 Figure 4.10: Standard deviation of the distance between cars as function of time Figure 4.11: Path of each of the about 3500 carsfigure 4.12: Maximum difference in distance of Figure through the simulation 4.11 Since VISSIM s vehicle record -function measures the travelled distance per timestep for each car, we can use this data to visualize the diffusion, Figure 4.11 shows the paths of all cars throughout the simulation. Calculating the maximum distance difference within this diffusion plot results in Figure This shows how the diffusion lineary increases. Furthermore, as wanted the effect of the very few ghost traffic jams is not visible. Figure 4.13: Histograms of the position of cars over time TU/e 14

16 Slicing the results of Figure 4.11 at fixed times gives us Figure 4.13, similar to [12]. This figure shows the histograms of the amount of cars as function of the position within the simulation. As expected due to the accelerations and decelerations, the distribution by the end is very similar to a gamma distribution. Furthermore, the distribution becomes wider over time due to the diffusion. Now that all the results of the simulations are discussed, we summarize these results in a conclusion in the next section, followed up by several recommendations. TU/e 15

17 5 Conclusion In Section 2, several equations for the approximation of the mean waiting time are discussed. These equations are relevant in the scenario of a fixed traffic light cycle and a poisson arrival rate, thus for an M/D/1-system. Of the formulas, Van de Broek claims to be the most accurate [6]. For the simulations of PTV VISSIM 7, it is validated that the inter-arrival time has an exponential distribution for both the exact and stochastic setting. Furthermore, in Section 4.2 we validated the deterministic distribution of the departure rate, meaning that an M/D/1-system is a reasonable assumption for traffic intersections with traffic signals cycling in a fixed sequence. This result also confirmed the choice of an M/D/1-system in the approximation formulas in Section 2. Since exact is used for the remaining simulations, the relevant coefficient of variation equals However, due to a feature of VISSIM, this is only the case when rounded off to 0.5s. The difference between the exact and stochastic setting is explained in Section 3.1 The departure rate during effective green time is determined for all three travel directions on an intersection. This is done for different direction-decisions at an intersection and for different speed limits. The results are displayed in Table 4.4. These departure rates need to be used for the equations given in Section 2. Furthermore, the effect of ghost traffic jams on the time between each car has been determined, this is done at a maximum arrival rate to generate as many traffic jams as possible. The traffic jams seem to have very minimal impact on the average time between each car. The average time between each car only slightly increases over a longer distance. However, the increase is too small to validate this hypothesis and there is also a high fluctuation of variance between each measured distance. More simulations are needed to confirm the hypothesis and to gain a more constant variance between each distance. Lastly, the effect of long travel time on the distance between each car is determined. While the average distance between each car remains the same over time, the standard deviation increases. Thus it is possible to visualise the diffusion of the cars over time, as represented in Figure Cutting the diffusion plot at several times give histograms of the position of the cars. At longer travel times, the histograms show more similarities with a gamma distribution. Now that we concluded this report, we discuss several recommendations in the next section. TU/e 16

18 6 Recommendations To improve the current results of Section 4.3, it is advisable to run additional simulations next to the current 30 simulations, making sure the variance will fluctuate less due to a more constant amount of ghost traffic jams for each distance. To make sure the delays are independent of eachother, the average delay of 30 simulations has to be calculated. This needs to be done 30 times to get a normal distribution of the average delay, which means a total of 900 simulations are necessary. While doing the simulations, it is important to keep the limits of PTV VISSIM 7 in mind, as explained in Section 3.2. Furthermore, it is also recommended to analyse for longer distances than 990 m. This is necessary to valide whether the mean time between each car increases or not over longer distances. Advised is to simulate at least up to 5 km. To make sure the simulations match real-life situations as much as possible, one should look at the accelerations after red light, since this influences the departure rates. The departure rates of Section 4.2 are in reality slighty lower, since this simulation only measures the steady-state during greentime. Furthermore, not only cars drive on the roads. Therefore, it is important to look at the effects when busses, trucks and agricultural vehicles are included. Another way to minimize the mean waiting time, is to look at the time of clearing the junction, a moment where all traffic-lights are red. One could minimize the clearing time, while making sure it is not to short due to safety [13]. TU/e 17

19 References [1] A. Bleijenberg. The attractiveness of car use. In Cars and Carbon, pages Springer, [2] PTV Planung Transport Verkehr, Haid-und-Neu-Str. 15, Karlsruhe Germany. PTV VISSIM 7 User Manual, [3] A. Miller and J. Alan. Settings for fixed-cycle traffic signals. Operational Research Quarterly, pages , [4] M. S. van den Broek, J. S. H. Van Leeuwaarden, I. J. Adan, and O. J. Boxma. Bounds and approximations for the fixed-cycle traffic-light queue. Transportation Science, 40(4), [5] F. V. Webster. Traffic signal settings. Road Research Technical Paper, 39, [6] M. S. van den Broek. Traffic signals. Optimizing and analyzing traffic control systems. Master s thesis, Technische universiteit Eindhoven, [7] A. J. H. Clayton. Road traffic calculations. Journal of the ICE, 16(7): , [8] M. Beckmann, C. B. McGuire, and C. B. Winsten. Studies in the economics of transportation. (226), [9] G. F. Newell. Queues for a fixed-cycle traffic light. The Annals of Mathematical Statistics, 31(3): , [10] R. Akçelik. Time-dependent expressions for delay, stop rate and queue length at traffic signals. Australian Road Research Board, [11] R. Wiedemann. Simulation des Verkehrsflusses. Schriftenreihe des Instituts für Verkehrswesen, 8, [12] E. Lefeber and D. Armbruster. Aggregate modeling of manufacturing systems. In K.G. Kempf, P. Keskinocak, and R. Uzsoy, editors, Planning Production and Inventories in the Extended Enterprise: A State of the Art Handbook, Volume 1, chapter 17, pages Springer International Series in Operations Research and Management Science, Volume 151, [13] M. Van Der Heijden, A. Van Harten, and M. Ebben. Waiting times at periodically switched one-way traffic lanes. Probability in the engineering and informational sciences, 15(04): , TU/e 18